On May 5, 8:56=A0am, "Dave L. Renfro" <renfr...@[EMAIL PROTECTED]
> wrote:
> michalc...@[EMAIL PROTECTED]
wrote:
> > You may be right about at least some kids getting
> > mislead by this technique. My main problems is that
> > wwhat you suggest is really not practical. For example
> > a common mistake kids make is accidently or mistakenly
> > cancelling part of a numerator or denominator as in
> > a^2/(a+1) --> a/1 --> a. There is no reversable step
> > that they can use to test. Or sometimes the reverse
> > step is just way too hard for the kid such as in
> > (2x^2 + x + 2) (x^2-2x+3) =3D 2*x^4-3*x^3+6*x^2-x+6.
> > Like I said, going over the probelm again is not
> > goos since people are prone to make the same mistake
> > twice. If you can come up with a better alternative
> > i would be grateful.
>
> I don't like your -3 test for two reasons. One reason
> is that the computations involved with plugging in -3
> are often not immediate, especially for 8th and 9th
> grade students.
True, but I was trading off the likelyhood of a false negative against
the difficulty of computation.
Another reason is that you're missing
> an op****tunity to reinforce some im****tant concepts
> (e.g. dividing causes exponents to subtract, etc.)
>
I am not so sure I agree with this, you'll have to expand this idea.
Anyway I teach the rules of exponest well enough that the kids usually
have little difficulty with them if they pay attention. The grades are
usually very high on that chapter.
> What you're looking at are things I called "safety
> checks" in my cl*****. The most im****tant two safety
> checks are plugging in 0 and looking at leading terms.
> Of course, you can't always use 0, so use 1 or -1,
> although now the process isn't so immediate.
>
> For example, I know (2x^2 + x + 2)(x^2 - 2x + 3)
> has a leading term of (2x^2)(x^2) =3D 2x^4 and a
> constant term of (2)(3) =3D 6, the latter obtained
> either by plugging in x =3D 0 or by knowing the
> lowest order term is obtained by appropriately
> combining the lowest order terms in the inputs.
>
> Also, I know (x^3 + 3x^2 - 4x - 12)/(x - 2)
> can't be x + 3 for two reasons. One is that
> the 'x' in x + 3 is not equal to x^3/x, and
> the other is that the '3' in x + 3 is not
> (-12)/(-2).
>
> A few years ago I taught 4 or 5 sections of (college)
> intermediate algebra one year, and I'd bet that 90%
> of the incorrect algebraic manipulation problems I
> found on tests could be seen immediately as incorrect
> by just applying the two safety checks above.
>
This is where our experiences differ. I find the most mistake in the
middle terms and in failing to group, cancel, distruute and factor
things completely. The highest and lowest order terms of the
expression are often correct.
> Now these aren't going to catch everything, but that's
> not the goal. It is not the goal to independently work
> the problem (especially when many of the students, and
> most of those in your target audience, can't work the
> problem correctly in just one way), but rather to have
> available 2-second "sight checks" that one can use as
> safety checks.
>
Yeah, you are probably right that they would benefit from being made
to look at their work briefly at this level.
> I note that one of your examples, a^2/(a+1) --> a/1 --> a,
> p***** both of my safety checks. However, this computation
> relies on two separate errors occurring together. The first
> is cancelling an 'a' without factoring an 'a' out in the
> denominator. The second error is in replacing the cancelled
> 'a' with 0 instead of 1. Now I don't doubt that this can
> happen (and I've almost certainly seen it, given how many
> papers I've graded in nearly 30 years), but as I've already
> indicated, the vast majority of algebraic errors can be
> detected with the two 2-second "sight checks" I mentioned.
In my experience that particular one happens with significant
frequency (albeit with more complex numbers). I don't think it is a
cognitive problem so much as a visual perception one where they have
not yet learned to look at a+1 as a single term when it is in a
denominator or numerator. Much of math is a perceptual skill as well
as a cognitive skill. A lot of people don't understand that, but that
is part of the underlying reason that when you introduce a new
technique it is im****tant to use simple numbers like single digit
intergers.
>
> Dave L. Renfro
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